Let $f(x)=3x-2$, and let $g(x)=f(f(f(f(x))))$.  If the domain of $g$ is $0\leq x\leq 2$, compute the range of $g$.
Solution: We iterate the function to find $g$:

\begin{align*}
f(f(x))&=3(3x-2)-2=9x-8\\
f(f(f(x)))&=3(9x-8)-2=27x-26\\
f(f(f(f(x))))&=3(27x-26)-2=81x-80
\end{align*}

This is an increasing, continuous function.  The minimum in the domain is at $0$, where it equals $-80$, and the maximum is at $2$, where it equals $-80+2(81)=82$.  It covers all values between these, so the range is $\boxed{-80\leq g(x)\leq 82}$.